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Chapter 2
X-Ray Emission from Black-Hole Binaries
M. Gilfanov
Abstract The properties of X-ray emission from accreting black holes are reviewed.
The contemporary observational picture and current status of theoretical understanding of accretion and formation of X-ray radiation in the vicinity of the compact
object are equally in the focus of this chapter. The emphasis is made primarily on
common properties and trends rather than on peculiarities of individual objects and
details of particular theoretical models. The chapter starts with discussion of the
geometry of the accretion flow, spectral components in X-ray emission, and black
hole spectral states. The prospects and diagnostic potential of X-ray polarimetry are
emphasized. Significant attention is paid to the discussion of variability of X-ray
emission in general and of different spectral components – emission of the accretion disk, Comptonized radiation, and reflected component. Correlations between
spectral and timing characteristics of X-ray emission are reviewed and discussed in
the context of theoretical models. Finally, a comparison with accreting neutron stars
is made.
2.1 Introduction
The gravitational energy of matter dissipated in the accretion flow around a compact
object of stellar mass is primarily converted into photons of X-ray wavelengths. The
lower limit on the characteristic temperature of the spectral energy distribution of the
emerging radiation can be estimated assuming the most radiatively efficient configuration – optically thick accretion flow. Taking into account that the size of the emitting region is r ∼ 10r g (r g is the gravitational radius) and assuming a black body
1/4
1/4
1/2
emission spectrum one obtains: kTbb ≈ L X /σ S B πr 2
≈ 1.4 L 38 /M10 keV. It
is interesting (and well known) that Tbb scales as directly proportional to M −1/2 .
This is confirmed very well by the measurements of the disk emission temperature
in stellar mass systems and around supermassive black holes in AGN. It is also
M. Gilfanov (B)
Max-Planck-Institute for Astrophysics, Garching, Germany; Space Research Institute, Moscow,
Russia, [email protected]
Gilfanov, M.: X-Ray Emission from Black-Hole Binaries. Lect. Notes Phys. 794, 17–51 (2010)
c Springer-Verlag Berlin Heidelberg 2010
DOI 10.1007/978-3-540-76937-8 2
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M. Gilfanov
illustrated, albeit less dramatically, by the comparison of the soft state spectra of
black holes and neutron stars, as discussed later in this chapter. The upper end of the
relevant temperature range is achieved in the limit of the optically thin emission. It is
not unreasonable to link it to the virial temperature of particles near the black hole,
kTvir = GMm/r ∝ mc2 /(r/r g ). Unlike the black body temperature this quantity
does not depend on the mass of the compact object, but does depend on the mass
of the particle m. For electrons Tvir ∼ 25(r/10r g )−1 keV and it is correspondingly
m p /m e = 1836 times higher for protons. Protons and ions are the main energy
reservoir in the accretion flow, but for all plausible mechanisms of spectral formation
it is the temperature of electrons that determines the spectral energy distribution of
the emerging radiation. The latter depends on the poorly constrained efficiency of
the energy exchange between electrons and protons in the plasma near the compact
object. The values of the electron temperature typically derived from the spectral fits
to the hard spectral component in accreting black holes, kTe ∼ 50 − 150 keV, are
comfortably within the range defined by the two virial temperatures. However, the
concrete value of kTe and its universality in a diverse sample of objects and broad
range of luminosity levels still remains unexplained from first principles, similar to
those used in the derivation of the temperature of the optically thick soft component.
Broadly speaking, significant part of, if not the entire diversity of the spectral
behavior observed in accreting black holes and neutron stars can be explained by
the changes in the proportions in which the gravitational energy of the accreting
matter is dissipated in the optically thick and optically thin parts of the accretion
flow. The particular mechanism driving these changes is however unknown – despite
significant progress in MHD simulations of the accretion disk achieved in recent
years (see Chap. 10) there is no acceptable global model of accretion onto a compact
object. To finish this introductory note, I will mention that non-thermal processes in
optically thin media (e.g., Comptonization on the non-thermal tail of the electron
distribution) may also contribute to the X-ray emission from black holes in some
spectral states.
2.2 Geometry and Spectral Components
The contributions of optically thick and optically thin emission mechanisms can be
easily identified in the observed spectra of accreting black holes as soft and hard
spectral components (Fig. 2.1). Depending on the spectral state of the source one of
these components may dominate the spectrum or they can coexist giving comparable
contribution to the total emission. The soft component is believed to originate in the
geometrically thin and optically thick accretion disk of the Shakura-Sunyaev type
[48]. To zeroth order approximation, its spectrum can be regarded as a superposition
of black body spectra with the temperatures and emitting areas distributed according
to the energy release and the temperature profile of the accretion disk. The simplest
example of this is the so-called multicolor disk black body model introduced by
[35]. Although valid for one particular inner torque boundary condition allowing
easy integration of the total flux, this model has been widely and efficiently used in
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19
Fig. 2.1 Energy spectra of a black hole Cyg X-1 (left) and a neutron star 4U 1705-44 (right) in the
soft and hard spectral states (left panel adopted from [18])
the era of more limited computer resources due to its simplicity, speed, and early
integration in the XSPEC spectral fitting package. To achieve a higher degree of
accuracy, one would need to consider more realistic inner boundary conditions,
deviations of the gravitational potential from the Newtonian, and to account for
such effects as distortion of the black body spectrum due to Thomson scatterings in
the upper layers and in the atmosphere of the disk, Doppler effect due to rotation of
the matter in the disk, etc. A number of models have been proposed to include these
effects [11, 49, 46, 4], many of them currently implemented in the XSPEC package.
The inadequacy of the optically thick emission mechanism in describing the
hard spectral component present in both spectra in Fig. 2.1 can be easily demonstrated. Indeed, the size of the emitting region required to achieve a luminosity
of log(L X ) ∼ 37 − 38 with black body temperature of ∼several tens of keV is
1/2
1/2
2
Rem ∼ L X /π σ S B T 4
∼ 20 L 37 /T30
m (T30 = Tbb /30 keV). Obviously this
is much smaller than the size of the region of main energy release near an accreting black hole, ∼50r g ∼ 500–1500 km. The bremsstrahlung emission from a ∼
uniform cloud of hot plasma with a filling factor close to unity cannot deliver the
required luminosity either, because the ∝ Ne2 density dependence of its emissivity
makes it very inefficient in the low density regime characteristic of the optically thin
accretion flow. Indeed, the optical depth of such a plasma cloud of size ∼10–100r g
and emission measure of Ne2 V ∼ 1059 cm−3 required to explain observed hard Xray luminosity of ∼1037 erg/s would greatly exceed unity. However, bremsstrahlung
emission may play a role in the advection-dominated accretion flow [37] in the low
Ṁ regime of quiescent state of accreting black holes (log(L X ) ≤ 32–33).
It has been understood early enough that Comptonization is the most plausible
process of formation of the hard spectral component [54, 55]. Thanks to its linear
dependence on the gas density, Comptonization of soft photons on hot electrons
in the vicinity of the compact object can efficiently radiate away the energy dissipated in the optically thin accretion flow and successfully explain the luminosity
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M. Gilfanov
and overall spectral energy distribution observed in the hard X-ray band. Moreover,
Comptonization models of varying degree of complexity do satisfactorily describe
the observed broadband energy spectra of black holes to the finest detail (e.g., [64]).
It is especially remarkable given the high quality of the X-ray data which became
available in the last decade from observations of recent and current X-ray observatories, such as Compton GRO, RossiXTE and Chandra, XMM and Swift.
The Comptonization site – cloud of hot (thermal or non-thermal) electrons is
often referred to as a “corona”. Although it is generally accepted that the Comptonizing corona has to be located in the close vicinity of the compact object, there
is currently no broad consensus on the detailed geometry of the region. The numerical simulations have not reached the degree of sophistication required to perform
full self-consistent global simulations of the accretion flow. Two physically motivated geometries are commonly considered: (i) the “sombrero” configuration (to
my knowledge first introduced in [40]) and (ii) the patchy, flaring corona above the
accretion disk.
One of the variants of the “sombrero” configuration is depicted in Fig. 2.2. It is
assumed that outside some truncation radius the accretion takes place predominantly
Fig. 2.2 The three main components of the X-ray emission from an accreting black hole (top) and
a plausible geometry of the accretion flow in the hard spectral state (bottom)
2 X-Ray Emission from Black-Hole Binaries
21
via the optically thick and geometrically thin accretion disk, whereas closer to
the compact object the accretion disk is transformed into a hot optically thin and
geometrically thick flow with the aspect ratio of H/R∼0.5–1. The soft (optically
thick) and hard (optically thin) spectral components are formed in the accretion
disk and the hot inner flow correspondingly. The value of the truncation radius
can be inferred from the observations. Although their interpretation is not unique
and unambiguous, the plausible range of values is between ∼3 and a few hundred
gravitational radii. There is no commonly accepted mechanism of truncation of the
disk and formation of the corona, with a number of plausible scenarios having been
investigated recently. Among the more promising ones is the evaporation of the
accretion disk under the effect of the heat conduction. It was initially suggested
to explain quiescent X-ray emission from cataclysmic variables [31] and was later
applied to the case of accretion onto black holes and neutron stars [32]. It not only
provides a physically motivated picture describing the formation of the corona and
the destruction of the optically thick disk but also correctly predicts the ordering
of spectral states vs. the mass accretion rate. Namely, it explains the fact that hard
spectra indicating prevalence of the hot optically thin flow are associated with lower
Ṁ values, whereas the optically thick disk appears to dominate the photon production in the accretion flow at higher Ṁ. As a historical side note I mention that in the
earlier years of X-ray astronomy the presence of the hot optically thin plasma in the
vicinity of the compact object was often associated with disk instabilities, therefore
such a behavior appeared puzzling to many astrophysicists in the view of the Ṁ
dependence of these instabilities.
Another geometrical configuration considered in the context of hard X-ray emission from black holes is a non-stationary and non-uniform (patchy) corona above
the optically thick accretion disk. This scenario has been largely inspired by the
suggestion by Galeev et al. [13], that a magnetic field amplified in the hot inner disk
by turbulence and differential rotation may reach the equipartition value and emerge
from the disk in the form of buoyant loop-like structures of solar type above its
surface. These structures may lead to the formation of a hot magnetically confined
structured corona similar to the solar corona, which may produce hard emission via
inverse Compton and bremsstrahlung mechanisms. The model could also explain
the faintness of the hard emission in the soft state as a result of efficient cooling of
plasma in the magnetic loops via inverse Compton effect due to increased flux of
soft photons at higher Ṁ.
Although the original Galeev, Rosner, and Vaiana paper was focused on the thermal emission from hot plasma confined in buoyant magnetic loops, the latter are a
plausible site of particle acceleration responsible for the non-thermal component in
the electron distribution. This is expected on theoretical grounds and is illustrated
very well observationally by the presence of the non-thermal emission component
in the spectra of solar flares (see e.g., [9] and references therein). On the other hand,
a power-law-shaped hard X-ray component is commonly observed in the spectra
of black hole candidates (Fig. 2.3). It reveals itself most graphically in the soft
spectral state but may be as well present in the hard state, along with the thermal
Comptonized spectrum (e.g., [33]). This power-law component has a photon index
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M. Gilfanov
Fig. 2.3 Broadband spectra of Cyg X-1 in the soft and hard spectral state based on the data of
BeppoSAX and Compton GRO missions. Adopted from [33]
of Γ ∼ 2–3 and, although it may extend into the several hundred keV–MeV range
(e.g., [58]), it is relatively unimportant energetically, contributing a small fraction to
the total radiation output of the black hole. This is in contrast with the hard spectral
component due to thermal Comptonization, observed in the hard spectral state which
accounts for the most of the source luminosity.
There have been attempts to explain the power-law component as a result of
Comptonization in a media with Maxwellian distribution of electrons of large temperature and small optical depth. This seems to be unlikely as in this case humps
due to individual scattering will be seen in the output spectrum, in contrast with the
smooth power-law spectra observed. This qualitative consideration is illustrated by
results of Monte-Carlo simulations shown in Fig. 2.4.
The sombrero configuration is often associated with a predominantly thermal
distribution of electrons, whereas the solar-like flares above the accretion disk may
be the site of electron acceleration producing non-thermal electron distributions and
power-law-like Comptonization spectra. Thermal Comptonization is believed to be
the main mechanism in the hard spectral state, whereas non-thermal Comptonization is probaly relevant in the soft state (Fig. 2.3). On the other hand, observations
often indicate presence of both thermal and non-thermal Comptonization components suggesting that the two types of corona may coexist (e.g. [23]).
Due to heating of the disk by Comptonized radiation from the corona and soft
photon feedback, the uniform thermal corona above the optically thick accretion
disk cannot explain observed hard spectra with photon index of Γ ∼1.5–2.0. It
has been first proposed by Sunyaev and Titarchuk [56] that the presence of cool
optically thick media – e.g., accretion disk or surface of the neutron star, in the
vicinity of the Comptonization region will affect the parameters of the latter and,
consequently, the shape of the outgoing Comptonized radiation. Indeed, some fraction of the Comptonized radiation will be returned to the accretion disk increasing its temperature and, consequently, the soft photon flux to the Comptonization
2 X-Ray Emission from Black-Hole Binaries
23
Fig. 2.4 Comptonization spectrum in the case of large temperature and small optical depth. Left
panel: kTbb = 100 eV, kTe = 300 keV, τ = 0.01, 0.05, 0.1 from the bottom to the top; right panel:
kTbb = 500 eV, kTe = 1 MeV, τ = 0.001, 0.01, 0.1. For comparison, the topmost spectrum on the
left panel shows case of τ = 1. The left-most peak in all spectra is made of seed photons which left
the Comptonization region without scatterings. The dashed line is a power law with a photon index
of Γ = 3. These spectra are the results of Monte-Carlo simulations assuming spherical geometry
region. This in turn will increase the cooling rate and will decrease the electron
temperature in the Comptonization region leading to softer and steeper spectra. I
will further illustrate it by the following simple quantitative consideration. In the
sandwich-like geometry, assuming moderate Thompson optical depth of the corona,
τT ∼ 1, a fraction f ∼ 1/2 of the Comptonized emission will be returned to the
accretion disk. Of this, a fraction of 1−α (α ∼ 0.2 – albedo) will be absorbed
and will contribute to the heating of the accretion disk, adding to its heating due
to the gravitational energy release. Ignoring the latter, the luminosity enhancement
factor in the Comptonization region, defined as a ratio of its total luminosity to the
luminosity of the seed photons (see more detailed discussion in Sect. 2.8.2) will be
A ≈ (1−α)−1 f −1 < 2.5. As well known [10], the luminosity enhancement factor is
intimately related to the Comptonization parameter y and the photon index Γ of the
Comptonized radiation. The above constrain on A implies Γ > 2.3 (see Fig. 2.15),
which is steeper than the hard state spectra typically observed in black holes. This
conclusion is confirmed by the full treatment of the Comptonization problem in
“sandwich” geometry [22]. In order to produce a harder Comptonized spectrum,
the value of the feedback coefficient f needs to be reduced. This is achieved, for
example, in a non-uniform, patchy and/or non-stationary corona. Another example
suggested by [5] involves bulk motion of the corona with mildly relativistic velocity
away from the disk reducing the feedback coefficient due to the aberration effect
(Sect. 2.8.2). A uniform stationary corona above the accretion disk can, in principle,
be responsible for the steep power-law component often detected in the soft state,
although a non-thermal electron distribution may be a more plausible explanation,
as discussed above. To conclude, I also note that considerations of a similar kind
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M. Gilfanov
involving the neutron star surface can explain the fact that the neutron star spectra
are typically softer than those of black holes (e.g., [56], Sect. 2.6).
2.3 Spectral States and Geometry
The existence of different spectral states is a distinct feature of accreting X-ray
sources, independently of the nature of the compact object (Fig. 2.1, 2.3). Although
their phenomenology is far richer, for the purpose of this chapter I will restrict
myself to the simple dichotomy between soft (high) and hard (low) spectral states
and refer to the next chapter of this book for a more detailed discussion. As
no global self-consistent theory/model of accretion exists, all theories explaining
spectral states have to retreat to qualitative considerations. These considerations,
although phenomenological in nature, usually are based on numerous observations
of black hole systems, simple theoretical arguments, and some simplified solutions
and simulations of the accretion problem. Described below is a plausible, although
neither unique nor unanimously accepted, scenario of this kind based on the “sombrero” geometry of the accretion flow. There are a number of cartoons and geometry
sketches, illustrating this and other scenarios which I will not repeat here and will
refer the interested reader to the original work, e.g., [68]
As obvious from Fig. 2.1, the spectral states phenomenon is related to the redistribution of the energy released in the optically thick and optically thin components
of the accretion flow. In the sombrero configuration one may associate spectral state
transitions with change of the disk truncation radius – the boundary between the
outer optically thick accretion disk and the inner optically thin hot flow.
In the soft (aka disk-dominated) spectral state, the optically thick accretion disk
extends close to the compact object, possibly to the last marginally stable Keplerian
orbit (r = 3r g for a Schwarzschild black hole), leaving no “room” for the hot optically thin flow. Therefore, the major fraction of the accretion energy is emitted in the
optically thick accretion disk giving rise to a soft spectrum of the multicolor black
body type. The magnetic activity at the disk surface may (or may not) produce a hard
power law-like tail due to non-thermal Comptonization in the corona. As discussed
in the previous section, this power-law component has a steep slope Γ ∼2–3 and is
relatively insignificant energetically.
In the hard (aka corona-dominated) spectral state, the accretion disk truncates
at a distance of ∼50–100r g or further from the compact object. The major fraction
of the gravitational energy is released in the hot inner flow. Comptonization of the
soft photons emitted by the accretion disk on the hot thermal electrons of the inner
flow leads to the formation of the hard spectrum of the shape characteristic for
unsaturated thermal Comptonization. The typical parameters in the Comptonization region – hot inner flow, inferred from observations are: electron temperature of
Te ∼ 100 keV and Thompson optical depth of τe ∼ 1. The significance of the soft
backbody-like emission from the optically thick disk as well as of the non-thermal
emission due to magnetic flares at its surface varies depending on the disk truncation
2 X-Ray Emission from Black-Hole Binaries
25
radius, increasing as the disk moves inward. There is evidence that both thermal and
non-thermal hard components may coexist in the hard state in the certain range of
the disk truncation radii, as suggested in [23].
2.4 Reflected Emission
After escaping the corona, a fraction of the Comptonized photons may be intercepted by the optically thick accretion disk. Part of the intercepted radiation will
be dissipated in the disk, via inverse Compton effect and photoabsorption on heavy
elements, contributing to its energy balance, the remaining part will be reflected due
to Compton scatterings [3]. The disk albedo depends on the photon wavelength and
on the chemical composition and ionization state of the disk material. Expressed in
terms of energy flux it is ∼0.1–0.2 in the case of neutral matter of solar abundance
[30]. Some fraction of the photobsorbed emission will be reemitted in the fluorescent lines of heavy elements and may also escape the disk. Thus, combined effects of
photoabsorption, fluorescence, and Compton scattering form the complex spectrum
of the reflected emission, consisting of a number of fluorescent lines and K-edges
of cosmically abundant elements superimposed on the broad Compton reflection
hump [3, 14] (Fig. 2.5). The peak energy of the latter depends on the shape of the
incident continuum and a typical spectrum of a black hole in the hard state is located
at ∼30 keV (Fig. 2.2).
Due to the dependence of the photoabsorption cross section, element abundances
and fluorescent yield on the atomic number, the strongest among the spectral features associated with heavy elements are the K-edge and fluorescent K-α line of
Fig. 2.5 Spectrum of emission reflected from an optically thick slab of neutral matter of solar
abundance. The solid line shows the spectrum of the incident power-law emission
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M. Gilfanov
iron. For reflection from the neutral media, the centroid of the iron line is at 6.4 keV
and its equivalent width computed with respect to the pure reflected continuum is
∼1 keV. For a distant observer, the reflected emission is diluted with the primary
Comptonized radiation and the thermal emission of the accretion disk, adding complexity to the observed spectra. It leads to the appearance of characteristic reflection
features in the spectra of X-ray binaries (Fig. 2.5) – the fluorescent Kα line from
iron, iron K-edge, and a broad Compton reflection hump at higher energies [3, 14].
Their shape is further modified, depending on the ionization state of the matter in
the disk [47], by strong gravity effects and Doppler and aberration effects due to
the Keplerian rotation of the disk, e.g., [12]. The observed line usually has nonzero intrinsic width and the K-edge is never sharp (smeared edge). Their amplitude
depends on the ionization state and on the solid angle of the reflector as seen from
the source of the primary radiation. This can change with the spectral state of the
source. The typical value of the iron line equivalent width is ∼50–300 eV.
The presence of these features makes the spectra deviate from a power-law shape,
expected for Comptonized radiation with parameters typical for black holes. As their
amplitude depends on the relative configurations of the corona and the accretion
disk and physical conditions in the disk, they have great diagnostics potential for
studying the accretion geometry in different spectral states. This is further discussed
in Sect. 2.8.
2.5 Polarization of X-Ray Emission
As the distribution of matter in the accretion flow is not spherically symmetric, one
may expect some degree of polarization of emerging X-ray radiation. In the conventional scenario of formation of X-ray radiation in the vicinity of a compact object
the polarization is caused by Thompson scatterings of photons on free electrons in
the accretion disk and hot corona and is predicted to be present at the moderate
level of ∼ several percent in all three main components of the X-ray emission:
thermal emission of the optically thick disk, Comptonized emission, and reflected
component. A considerably larger degree of polarization may be expected in some
alternative scenarios, for example, in some versions of the jet scenario, discussed
later in this book.
The degree of polarization of the Comptonized emission depends strongly on
the geometry of the corona, the location of the sources of the soft seed radiation,
and the viewing angle [57, 63]. As calculated by Sunyaev and Titarchuk [57], for
the case of Comptonization in the disk it lies in the range between 0 and ∼12%,
depending on the optical depth of the corona and viewing angle, i.e., it can slightly
exceed the maximum value of 11.7% for a pure-scattering semi-infinite atmosphere.
Although the disk geometry of the corona is unlikely, this result suggests that a
moderate degree of polarization of the Comptonized continuum, of the order of
∼ few percent, may be expected.
Obviously, polarization should also be expected for the reflected emission from
the accretion disk. The pure reflected component can be polarized to ∼30% [29].
2 X-Ray Emission from Black-Hole Binaries
27
The degree of polarization drops to ∼ few percent when it is diluted with the Comptonized radiation and the thermal emission from the disk. The degree of polarization
is a strong function of photon energy – it is low at low energy <10 keV and reaches
its maximum of ∼5% (for a 60◦ viewing angle) at ∼30–50 keV, where the contribution of the Compton-reflected continuum to the overall spectrum is maximal [29].
The thermal emission generated inside the optically thick accretion disk is initially unpolarized but attains polarization as a result of Thompson scatterings in the
atmosphere of the accretion disk [8, 27]. Calculations [27] show that the degree
of polarization varies from 0% (face-on disk) to ∼5% (edge-on). The polarization
degree depends on the photon energy, being largest in the Wien part of the thermal
spectrum, where scatterings play the most important role. A remarkable property of
polarized radiation from the vicinity of a compact object is that due to relativistic
effects the polarization angle also becomes dependent on the photon energy [8, 27].
This is in contrast with the classical approximation, where symmetry considerations
require that the polarization direction is coaligned with the minor or the major axis
of the disk projection on the plane of the sky. The amplitude and shape of these
dependences are sensitive to the disk inclination and the spin of the black hole. In
combination with spectral information, this can be used to resolve the degeneracy
between the black hole mass, spin and, disk inclination [27].
Polarization measurements are yet an unexplored area of high energy astrophysics and have great diagnostics potential. With the new generation of polarimetric detectors, X-ray polarimetry will become a powerful tool to study the geometry
of the accretion flow and the properties of the compact object in accreting systems.
2.6 Variability
Variability of X-ray emission is a common and well-known property of X-ray binaries. The amplitude of flux variations depends on the timescale and the photon
energy and can be as large as ∼20–30% fractional rms. The dependence of the variability amplitude on timescale is conventionally characterized by the power density
spectrum, which is a square of the Fourier amplitudes of the time series, renormalized in order to give the answer in desired units, for example (fractional rms)2 /Hz.
Power density spectra of X-ray sources often reveal a number of rather narrow
features of various widths superimposed on a broadband continuum of aperiodic
variations, suggesting that both resonances of various degrees of coherence as well
as stochastic processes of aperiodic nature contribute to the observed flux variations.
Some of the narrow features can be clearly associated with the spin frequency of the
neutron star or orbital or precession frequency of the binary. The nature of others, broadly referred to as quasi-periodic oscillations (QPO), is poorly understood.
The models range from the beat frequency model proposed by Alpar and Shaham
[2] soon after the discovery of the first QPOs in neutron stars, and the relativistic
precession model [53] relating QPO frequencies with fundamental frequencies of
Keplerian orbits in strong gravity to models employing global oscillation modes in
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M. Gilfanov
the accretion disk [1, 60]. QPOs were first discovered in neutron star systems, but
are also commonly observed in accreting black holes. Their rich phenomenology is
well documented elsewhere (e.g., [61]) and is not discussed in this chapter, which
will focus on the aperiodic variability continuum.
I will conclude these introductory remarks with the following side note. The
interpretation of the energy spectra of celestial X-ray sources has been greatly facilitated by the fact that a number of simple physical concepts could be employed in
a straightforward manner, such as black body emission, Comptonization, photoabsorption, and fluorescence. No equivalent concepts are easily available to help in
interpreting the power density spectra. This may explain why our understanding of
the power density spectra lags significantly behind our understanding of the energy
spectra. Indeed, spectral analysis can rely on a number of physically motivated
and elaborate models which successfully describe high quality data from modern
observatories. The interpretation of the power density spectra, on the contrary, has
just started to advance beyond “numerology” and simple association of QPO frequencies with characteristic frequencies of test particles in the (strong) gravitational
field. However, it is obvious that timing information (along with the polarization
measurements) presents a completely different dimension which has to be taken
into account in validating any model of formation of radiation in the vicinity of a
compact object.
2.6.1 Propagation of Ṁ Fluctuations in the Accretion Disk
The remarkable characteristics of aperiodic variability is its breadth in the frequency
domain. As illustrated by the power density spectra of Cyg X-1 and Cyg X-2
shown in Fig. 2.6, flux variations in the frequency range from log f < −8 to
log f ∼ 2 − 3 are present. This suggests that variations of the mass accretion
rate Ṁ on an extremely broad range of timescales are present in the innermost
region of the accretion flow, r ≤ 50–100 r g , where X-ray emission is produced.
It is to be compared with characteristic timescales in this region which are limited by two extremes – the dynamical timescale that is of the order of the Kep3/2
lerian orbital time t K ≈ 0.3 (M/10M ) r/50r g
s and the viscous timescale
−1
−2
−2
−1
−1
tvisc ∼ α (h/r ) Ω K = (2π α) (h/r ) t K (α is viscosity parameter, h – the
disk thickness). The latter is in the range between tvisc ∼ 104 t K ∼ 103 s for a gas
pressure-dominated Shakura-Sunyaev disk (h/r ∼ 10−2 ) and tvisc ∼ 10 t K ∼ 1–10 s
for a thicker hot flow with the aspect ratio of h/r ∼ 1/3.
From the point of view of characteristic timescales, the high-frequency variations could potentially be produced in the vicinity of the compact object. Longer
timescales, on the contrary, exceed by many orders of magnitude the longest
timescales in the region of the main energy release and cannot be generated there.
The low frequency Ṁ variations have to be generated in the outer parts of the
accretion flow and be propagated to the region of the main energy release where
they are converted into variations of the X-ray flux. The power spectra shown in
2 X-Ray Emission from Black-Hole Binaries
29
Fig. 2.6 Broad band power density spectra of Cyg X-1 and Cyg X-2. The spectra are obtained
from non-simultaneous RXTE/ASM, EXOSAT/ME, and RXTE/PCA data. In the left panel, the
open circles at log( f ) > −4 (red in the color version of the plot) show the 1996 high state PDS,
the filled circles (blue) are hard spectral state PDS. The solid lines extending through entire plots
in the left and right panels show a Pν ∝ ν −1.3 power law; normalization is the same on both
panels. The second power law in the left panel giving a better approximation to the soft state data
of Cyg X-1 is Pν ∝ ν −1 power law. The very high frequency data log( f ) > 1 are from [43]
Fig. 2.6 maintain the same power-law shape over a broad frequency range, suggesting that the same physical mechanism is responsible for flux variations at all
frequencies. The same slope and normalization of the power-law component in the
power density spectra of different sources – black holes and neutron stars – suggests
that this mechanism is a property of the accretion disk and does not depend on the
nature of the compact object. A plausible candidate for such mechanism may be
viscosity fluctuations caused for example, by MHD turbulence in the accretion disk,
as suggested in [26]. Viscosity fluctuations lead to variations of the mass accretion
rate which after propagating into the innermost region of the accretion flow are
transformed into variations of the X-ray flux.
However, because of the diffusive nature of disc accretion [48], fluctuations of
mass accretion rate on timescales much shorter than the diffusion timescale will not
be propagated inward, but will be damped close to the radius at which they originated. As demonstrated in [7], the amplitude of the fluctuations on the timescale
τ
√
will be significantly suppressed at the characteristic length scale Δr/r ∼ τ/tvisc .
As the viscous timescale depends quadratically on the disk thickness, tvisc ∼
(h/r )−2 Ω K−1 , fluctuations on the dynamical τ ∼ td ∼ Ω K−1 or thermal τ ∼ tth ∼
(αΩ K )−1 timescales will be damped in the thin disk with h/r 1 after traveling a
small distance in the radial direction, Δr/r ∼ h/r , and will never reach the region
of the main energy release. The inner region itself cannot generate Ṁ fluctuations
on the dynamical or thermal timescales either. Indeed, the coherence length for
fluctuations on a timescale τ ∼ Ω K−1 is Δr/r ∼ h/r ∼ 10−2 for the gas pressure
supported Shakura-Sunyaev disk. Therefore N ∼ r/Δr ∼ 100 independent annuli
30
M. Gilfanov
regions will contribute to the observed flux, their variations being uncorrelated. The
contribution of each region to the total flux is small, ∼10−2 , and alone cannot cause
significant variability of the total flux. Furthermore, in their combined
emission
√
uncorrelated fluctuations will be suppressed by a factor of ∼ 1/ N ∼ 10−1 due to
the averaging effect. Thus, for the geometrically thin disc, fluctuations of viscosity
(or mass accretion rate) on the dynamical or thermal timescales will not contribute
significantly to observed variability of the X-ray flux. In order to lead to significant
modulation of the X-ray flux, viscosity or accretion rate fluctuations have to propagat inward significant radial distance and cause fluctuations of mass accretion rate in
a significant range of smaller radii, including the region of the main energy release.
This is only possible for fluctuations on timescales equal or longer than the viscous
time of the disk at the radius, where fluctuations are “inserted” into the accretion
flow [26, 7].
Thus, the standard Shakura-Sunyaev disk plays the role of a low-pass filter, at
any given radius r suppressing Ṁ variations on timescales shorter than the local
viscous time tvisc (r ). The viscous time is, on the other hand, the longest timescale
of the accretion flow; therefore, no significant fluctuations at longer timescales can
be produced at any given radius. Hence, a radius r contributes to Ṁ and X-ray flux
variations predominantly at a frequency f ∼ tvisc (r )−1 [26, 7]. The broad range
of variability timescales observed in the X-ray power density spectra (Fig. 2.6) is
explained by the broad range of radii at which viscosity fluctuations are generated.
If viscosity fluctuations at all radii have the same relative amplitude, a power-law
spectrum Pν ∝ ν −1 will naturally appear [26], in qualitative agreement with observations (Fig. 2.6). The picture of inward-propagating fluctuations outlined above
also successfully explains the observed linear relation between rms amplitude of
aperiodic variability and total X-ray flux in black holes [62] and the nearly logarithmic dependence of the time lag between time series in different energy bands on the
photon energy [25].
2.6.2 Very Low Frequency Break and Accretion Disk Corona
Owing to the finite size of the accretion disk, the longest timescale in the disk is
restricted by the viscous time on its outer boundary tvisc (Rd ). Below this frequency,
X-ray flux variations are uncorrelated; therefore, the power density spectrum should
become flat at f ≤ tvisc (Rd )−1 . This explains the low frequency break clearly seen
in the power spectrum of Cyg X-2 (Fig. 2.6). If there are several components in the
accretion flow, for example, a geometrically thin disk and a diffuse corona above
it, several breaks can appear in the power spectrum at frequencies corresponding to
the inverse viscous timescale of each component. It can be easily demonstrated [21]
that the break frequency is related to the orbital frequency of the binary via
f break = 3π α(1 + q)−1/2 (h/r )2 (Rd /a)−3/2 f orb
2 X-Ray Emission from Black-Hole Binaries
31
0.3
0.1
H/R=0.02-0.04
−1
Fig. 2.7 Relation between the break frequency (∼ tvisc
at the outer edge of the accretion disk)
and the orbital frequency of the binary system. The shaded area shows the dependence f visc vs.
f orb expected for the standard gas pressure supported Shakura-Sunyaev disk (h/r ∼ few × 10−2 ).
Straight solid and dashed lines are predictions for larger values of the disk thickness h/r , as indicated by the numbers on the plot. The two points for Cir X-1 correspond to the original data (open
circle) and after correction for the eccentricity of the binary orbit (filled circle). Adopted from [21]
For low-mass X-ray binaries (Roche-lobe filling systems) this becomes f break ∝
(h/r )2 f orb . Such very low frequency breaks indeed are observed in a statistically
representative sample of low mass X-ray binaries, and the break frequency is
inversely proportional to the orbital period of the binary (Fig. 2.7, [21]).
However, measured values of the break frequency imply that the viscous time of
the accretion flow is a factor of ≥10 shorter than predicted by the standard theory of
accretion disks (cf. shaded area in Fig. 2.7). This suggests that significant fraction
of the accretion Ṁ occurs through the geometrically thicker coronal flow above the
standard thin disk. Note that the existence of the Shakura-Sunyaev disk underneath
the coronal flow is required by optical and UV observations of LMXBs indicating
the presence of the optically thick media of extent comparable with the Roche-lobe
size [21]. The aspect ratio of the coronal flow implied by the f br / f orb measurements,
h/r ∼ 0.1, corresponds to a gas temperature of T ∼ 0.01 Tvir . The corona has
moderate optical depth in the radial direction, τT ∼ 1, and contains ≤10% of the
total mass of the accreting matter (but the fraction of Ṁ is much larger, probably
∼0.5). These estimates of temperature and density of the corona are in quantitative
agreement with the parameters inferred by the X-ray spectroscopic observations by
Chandra and XMM-Newton of complex absorption/emission features in LMXBs
with large inclination angle.
In this picture, the red noise (power-law) component of the observed variability
of the X-ray flux is defined by the viscosity and/or Ṁ fluctuations generated in
the coronal flow rather than in the geometrically thin disk. Fluctuations produced
in the standard Shakura-Sunyaev disk have much smaller amplitude due to a ∼10
times longer viscous timescale (see discussion in Sect. 2.6.1) and do not contribute
32
M. Gilfanov
significantly to the observed variability of X-ray flux. This is further supported by
the lack of variability in the soft black body component in Cyg X-1, as discussed
below.
2.6.3 High and Very High Frequencies
In addition to the power-law component of red noise dominating in the larger part
of the low frequency range, power density spectra usually have significant excess of
power in the high frequency end, f ≥ 10−2 − 10−1 Hz (Fig. 2.6). This excess power
is often referred to as a “band-limited noise,” due its limited scope in frequency. It
is this part of the power spectrum that is the subject of investigation in the majority
of “standard” timing analysis projects, for example, in the ones based on a typical
RXTE/PCA observation.
In black holes, this component is most prominent in the hard spectral state and
usually disappears in the soft state (Fig. 2.6, 2.8). Plotted in the ν Pν units, it shows
several broad humps, separated by a ∼0.5–1 decade in frequency (Fig. 2.8). The
centroid frequencies of these bumps vary with time but they usually do correlate
with each other [16, 65]. It has been proposed that they may be identified with the
(much more narrow) QPO peaks observed in the power spectra of accreting neutron
stars. When the proper identifications are made, black hole and neutron star systems
appear to follow the same global relation between QPO frequencies [41, 65, 38, 6].
This suggests that the features on the power density spectra and their frequencies
are the property of the accretion disk, their existence being unaffected by the nature
of the compact object.
Based on the amplitude and timescale considerations, it is plausible to link the
origin of the band-limited noise component to the inner hot optically thin flow in
Fig. 2.8 Power density spectra of Cyg X-1 in the soft and hard spectral states. The right hand
panel shows the very high frequency part of the power density spectrum. The power is shown in
units of ν Pν (left) and ν 2 Pν (right). Adopted from [18] and [43] respectively
2 X-Ray Emission from Black-Hole Binaries
33
the region of the main energy release. For a 10M black hole, the characteristic
time scales in this region (r ∼ 100r g ) are: t K ∼ 1 s and tvisc ∼ 10 s, the latter was
computed assuming α ∼ 0.2 and h/r ∼ 0.2. This suggest that the two main peaks
observed in the ν Pν spectra may (or may not) be related to the dynamical and viscous timescales at the inner edge of the truncated accretion disk and their origin may
be linked to the interaction between the geometrically thin outer accretion disk and
the optically thin hot inner flow. Note that what appears to be the lower frequency
hump in the ν Pν plot is in fact a break, below which the power density distribution
is flat, down to the frequency where the red noise component becomes dominant
(Fig. 2.6). As a flat power density distribution means lack of correlation between
events on the corresponding timescales, we can draw an analogy with the very low
frequency breaks associated with the viscous timescale at the outer edge of the disk
(Sect. 2.6.2). This further supports the association of the low frequency hump in the
ν Pν plot, Fig. 2.8 (= break in the power density spectrum) with the viscous timescale
at the outer boundary of the hot inner flow. One can estimate the truncation radius
of the geometrically thin disk (= outer radius of the hot inner flow in the sombrero
configuration, Sect. 2.2) equating the viscous time with the inverse break frequency,
f br−1 ∼5 s: rtr ∼102r g assuming α = 0.2 and the aspect ratio of the hot inner flow
h/r = 1/3. This number is in a good agreement with the disk truncation radius
inferred by other measurements, e.g., derived from the variability of the reflected
component ([18], Sect. 2.7).
The right panel in Fig. 2.8 also demonstrates that there is considerable power at
very high frequencies, at the level of ν Pν ∼ 1 − 3% at f ∼ 102 Hz. The millisecond
range includes a number of important characteristic timescales in the vicinity of the
compact object. The Keplerian frequency at the last marginally stable orbit for a
non-rotating black hole is f K ∼ 220 (M/10M )−1 Hz. The relativistic precession
frequencies associated with Keplerian motion in the vicinity of the compact object
are also in this range [53]. The typical value of the sound crossing time in the gas
pressure-supported thin disk corresponds to the frequency of a few hundred Hz. The
light-crossing time for the region of ∼10r g corresponds to the frequency of ∼1 kHz.
However, the observed power spectra break somewhere between f ∼ 20 Hz (soft
state) and f ∼ 50–100 Hz (hard state) and do not exhibit any detectable features
beyond these frequencies. The upper limit on the fractional rms of aperiodic variability in the 500–1000 Hz range is ≈1%. Similar upper limit can be placed on the
rms amplitude of narrow features anywhere in this range, ≈0.9% assuming width
ν/Δν = 20 (all upper limits at 95% confidence) [43]. This suggests that none of the
above processes leads to a significant variability and, in particular, no resonances are
present at their characteristic timescales. The nature of the high frequency breaks is
unclear. It may be plausible to associate them with the properties of the emission
mechanism of the hard component, for example, with the details and timescales
of heating and cooling of electrons in the Comptonization region. Obviously, these
properties are different for the hard component in the soft and hard states as indicated by the difference in the high frequency power density spectra shown in the
right panel in Fig. 2.8.
34
M. Gilfanov
2.6.4 (Lack of) Variability in the Disk Emission
The energy spectra of black holes in the soft state often have two spectral components, a soft black body-like component emitted by the optically thick accretion
disk and a hard power-law component of thermal or non-thermal origin formed
in the optically thin media (Sect. 2.3, Figs. 2.1,2.3). The study of the variability
properties of these two components in black hole X-ray novae in the early 1990s
with the Ginga and GRANAT observatories led to the conclusion that most of the
variability of the X-ray emission in the soft state is usually associated with the hard
spectral component (e.g., [36, 58, 15]), the thermal disk emission generally being
much more stable. This conclusion has been later confirmed by a more rigorous and
quantitative analysis [7, 18, 20] whose main results are summarized below.
The energy dependence of the variability can be characterized by the frequencyresolved energy spectrum [44]. It is defined as a set of Fourier amplitudes computed
from light curves in different energy channels. The Fourier amplitudes are integrated
over the frequency range of interest and expressed in the units of flux. Its advantages
over simple energy-dependent fractional rms are the possibility to use conventional
(i.e., response folded) spectral approximations and to compare its shape with shapes
of various spectral components present in the average energy spectrum of the source.
However, it cannot be always regarded as the energy spectrum of the variable component. For this to be possible, certain conditions must be fulfilled [20], for example,
(i) independence of the shape of the frequency-resolved spectrum on the Fourier
frequency and (ii) the absence of time lags between flux variations at different energies. If these two conditions are satisfied, the spectral variability can be represented
as F(E, t) = S0 (E) + I (t)S(E).
These conditions are fulfilled in the soft state of Cyg X-1 (Fig. 2.9): therefore,
the Fourier frequency-resolved spectrum represents the energy spectrum of the variable part of the X-ray emission. Remarkably, it coincides with the average source
spectrum at E ≥ 7 − 8 keV where the contribution of the disk emission becomes
small (Fig. 2.10). Moreover, source light curves in different energy channels allow
a linear decomposition in the form F(E, t) = A(E) + I (t)B(E) [7]. The constant
part of the source emission A(E) coincides with the spectrum of the accretion disk,
while the spectrum of the variable part B(E) coincides with the frequency-resolved
spectrum and can be described by the Comptonization model (Fig. 2.10). Thus,
on timescales from ∼100 ms to ∼500 s (at least) the source variability is due to
variations in the flux of the hard component, whose shape is kept constant in the
course of these variations, the amplitude of variability of the disk emission being
significantly smaller.
This can be explained in the model of inward-propagating fluctuations outlined
in Sect. 2.6.1. The hard component is associated with the optically thin hot flow
whose aspect ratio must be comparable to unity, h/r ∼ 0.1 − 1. Therefore, the
viscous timescale in the hot flow is ∼102 –104 times shorter than in the geometrically thin disk from which the soft component originates (cf. Sect. 2.6.2). This
makes the geometrically thick flow more “transparent” for the high frequency perturbations. Depending on the particular value of h/r and of the α-parameter, the
2 X-Ray Emission from Black-Hole Binaries
35
Fig. 2.9 Frequency-resolved spectra of Cyg X-1 in the soft state (June 16–18, 1996). The numbers
in the upper-right corner of each panel indicate the median frequency. In each panel, the upper histogram shows the average spectrum, the lower histogram shows the frequency-resolved spectrum
at low frequencies, 0.002–0.033 Hz (the same data as in the left panel of Fig. 2.10). The spectra are
plotted as ratio to a power-law spectrum with photon index Γ = 2.5 and low energy absorption
N H = 6 · 1021 cm−2
viscosity or Ṁ fluctuations on the thermal and even dynamical timescales will be
propagated inward without significant damping and will modulate the accretion rate
at all smaller radii, including the region where X-ray emission is formed, leading
to a significant modulation of X-ray flux. This is not the case for the geometrically
thin disk where the viscous timescale is ∼ 103 –104 times longer than the dynamical and thermal timescales; therefore, high frequency perturbations will be damped
(Sect. 2.6.1). It is plausible to expect that perturbations on the thermal and dynamical
timescales have larger amplitude than perturbations on the viscous timescales, thus
explaining the significantly smaller variability amplitude of the disk emission.
The lack of variability of the disk emission and its interpretation are consistent
with the conclusion of the Sect. 2.6.2 made from completely independent arguments.
36
M. Gilfanov
Fig. 2.10 Left: Average and frequency-resolved spectrum of Cyg X-1 in the 1996 soft state. Right:
Spectra of “constant” (open circles – A(E)) and “variable” (solid circles – B(E)) components
derived from the linear fits of the correlation between count rate in different channels. The normalization of the “variable” component B(E) is arbitrary. For comparison, the light grey (yellow
in the color version) curve shows the spectrum of a multicolor disk emission with a characteristic
temperature of 0.5 keV. The two upper spectra (solid squares) were averaged over the periods
of time when the count rate above 9 keV was high and low, respectively. The dark grey (green)
lines show that these spectra can be reasonably well (within 10–15%) approximated by a model
M(E) = A(E) + I ∗ B(E) consisting of the stable and variable spectral components where I (the
normalization of the variable component) is the only free parameter. The right panel is adopted
from [7]
Namely, based on the location of the low-frequency break in the power density spectra of neutron stars we concluded that the bulk of variability seen at low frequencies
originates in the optically thin coronal flow with an aspect ratio of h/r ∼ 0.1, rather
than in the underlying geometrically thin disk.
Finally, it should be noted that variations of the soft component may also arise
from the variations of the disk truncation radius. Such variations may be absent
if the geometrically thin disk extends to the last marginally stable orbit (3r g for
a Schwarzschild black hole) but appear when the disk truncation radius is larger.
Indeed, detailed study of the 1996 soft state of Cyg X-1 used as an example here
demonstrated that on a number of occasions, mostly at the beginning and in the end
of soft state episodes, the soft component was strongly variable. Notably, the power
density spectrum during these periods had a complex shape, significantly different
form the simple power law shown in Figs. 2.6, 2.8.
2.7 Variability of the Reflected Emission
The reflected component (Sect. 2.4) arises from the reprocessing of the Comptonized emission in the accretion disk (Fig. 2.2), therefore, it should be expected to
show some degree of variability as the Comptonized radiation is strongly
2 X-Ray Emission from Black-Hole Binaries
37
Fig. 2.11 The two effects defining the response of the reflected emission to variations of the Comptonized emission are the finite light travel time τd from the hot inner flow to the accretion disk and
the finite size of the accretion disk itself, Δrd /c ∼ Δτd (left panel). While the first factor introduces
a time delay between variations of the reflected and Comptonized components, the finite size of the
disk itself leads to the suppression of high frequency variations in the reflected emission. The right
panel shows the Green function of the time response of the geometrically thin disk around a 10M
black hole (from [18]). The numbers at the curves mark the disk truncation radius and inclination
variable. The characteristic times of absorption/emission processes in the accretion
disk are negligibly small; therefore, the main factors defining the response of the
reflected flux to variations of the Comptonized radiation are related to the light
travel times (Fig. 2.11). Namely, they are (i) the finite light travel time from the
source of primary radiation to the reflector, i.e., from the Comptonization region to
the accretion disk and (ii) the finite size of the reflector. The first will introduce a
time delay between variations of the reflected and Comptonized
components. The
amplitude of this delay is τd ∼ rd /c ∼ 10 rd /100r g (M/10M ) ms. The finite
size of the disk itself will lead to suppression of the high frequency variations in
the reflected component – the accretion disk acts as a low-pass filter. It seems to
be possible to estimate the cut-off frequency from the size of the accretion disk
making the main contribution to the reflected flux, Δrd ∼ rd , thus leading to
f cut ∼ Δτd−1 ∼ (Δrd /c)−1 ∼ 100 Hz, i.e., beyond the frequency range of the bulk of
observed variability (Fig. 2.8). However, calculations of the transfer function [18]
show that at the frequency f cut ∼ Δτd−1 the variability signal is suppressed by a
significant factor of ∼10–20, whereas a noticeable suppression of variability by a
factor of ∼2 or more occurs at frequencies ∼10 times lower. As these effects directly
depend on the mutual location and the geometry of the Comptonization region and
the accretion disk, their observation is a powerful tool in studying the geometry of
the accretion flow.
As suggested in [44, 18], the variability of the reflected emission can be studied
using the methods of frequency-resolved spectroscopy (Sect. 2.6.4). This method
is based on the analysis of Fourier amplitudes and therefore washes out phase
38
M. Gilfanov
Fig. 2.12 Suppression of high frequency variations (left) and time delay (right) in the reflected
emission. The left panel shows the equivalent width of the iron fluorescent line (= ratio of the
fractional rms of the iron line flux to that of the underlying continuum) vs. Fourier frequency. The
model curves are for an isotropic point source at the height h = 10r g on the axis of a flat disk
with inner truncation radius of 10, 100 and 1000r g (assuming a 10M black hole) and with an
inclination angle of 50◦ . A narrow line was assumed in calculations. The right panel shows the
cross-correlation function of the iron line emission and the power-law continuum. Positive lags
mean delay of the line emission with respect to the continuum
information, precluding the study of time delay effects. However, it provides a
convenient way to explore the frequency dependence of the variability of the iron
line flux and to compare it with that of the continuum emission which is dominated by the Comptonized radiation. Using this method, [44, 18] showed that in the
soft spectral state, variations of the reflected component have the same frequency
dependence of the rms amplitude as the Comptonized emission up to frequencies
∼30 Hz (Fig. 2.12). This would be expected if, for instance, the reflected flux was
reproducing, with a flat response, the variations of the primary radiation down to
timescales of ∼30–50 ms. The sensitivity of their analysis was insufficient to study
shorter timescales. In the hard spectral state, on the contrary, the variability of the
reflected flux is significantly suppressed in comparison with the direct emission
on timescales shorter than ∼0.5–1 s. These findings are to be compared with the
predictions of the simple model for the time response of the disk to variations of
the primary emission. Assuming that suppression of the short-term variability of
the reflected emission is caused by the finite light-crossing time of the accretion
disk, one can estimate the truncation radius of the accretion disc, rd ∼ 100r g in the
hard spectral state and rd ≤ 10r g in the soft spectral state (Fig. 2.12). This agrees
well with the interpretation of the spectral states in black holes in the “sombrero”
geometry of the accretion flow (Sect. 2.2).
In order to study the time delay in the reflected emission, one would need to
separate the reflected component from the main emission in time-resolved energy
spectra. This can be done most easily for the fluorescent iron line, while the reflection continuum is more difficult to separate due to its breadth in the energy domain.
2 X-Ray Emission from Black-Hole Binaries
39
An attempt to perform such an analysis based on RXTE/PCA data of Cyg X-1 in the
hard state is presented in the right panel in Fig. 2.12. In each bin of the light curve
with 16 ms resolution, the spectrum in the 3–20 keV band was linearly decomposed
into power-law component, reflected continuum, and 6.4 keV iron line emission.
The cross-correlation of the light curves of the iron line flux and the power-law
component is shown in Fig. 2.12. The cross-correlation function shows a clear
asymmetry, suggesting a time delay of the order of ∼10–15 ms. The amplitude of
the possible time delay corresponds to a disk truncation radius of ∼100r g , in good
agreement with the number obtained from the analysis of high-frequency variations
in the iron line flux (left panel in Fig. 2.12) and also with the numbers tentatively
suggested by spectral analysis.
The results shown in Fig. 2.12 seem to suggest a rather consistent picture and
favor to the “sombrero”-type configuration of the accretion flow, with the spectral
state transition being related to a change of the disk truncation radius. However, a
caveat is in order. While the rms amplitude behavior of the iron line emission is a
rather robust observational result, the search of the time delay of the reflected component requires separation of line and continuum emission components. On ∼10 ms
timescales, this cannot be done through direct spectral fitting because of insufficient
statistics, even with the large collecting area of the PCA instrument aboard RXTE
and requires more sophisticated data analysis techniques, for example, the one used
to produce the cross-correlation function shown in Fig. 2.12. Second, although the
interpretation of these results in terms of finite light travel times is the most simple
and straightforward, alternative scenarios are also possible as discussed in length in
[18]. Nevertheless, the former interpretation is, in my view, the simplest and most
attractive one and is further supported by the results of the spectral analysis, as
described in the next section.
2.8 R − Γ and Other Correlations
Observations show that spectral and timing parameters of accreting black holes
often change in a correlated way, e.g., [16, 42, 39]. One of the most significant
correlations is the one between the photon index of the Comptonized spectrum, the
amplitude of the reflected component, and the characteristic frequencies of aperiodic
variability (Figs. 2.13, 2.16). The correlation between spectral slope and reflection
amplitude is also known as R − Γ correlation [16, 66]. Its importance is further
amplified by the fact that it is also valid for supermassive black holes (Fig. 2.14)
[67].
2.8.1 R − Γ Correlation
The spectrum formed by the unsaturated Comptonization of low frequency seed
photons with characteristic temperature Tbb on hot electrons with temperature Te
has a nearly power-law shape in the energy range from ∼3kTbb to ∼kTe [55]. For
40
M. Gilfanov
Fig. 2.13 The R − Γ correlations between the photon index Γ of the Comptonized radiation and
the relative amplitude of the reflected component R in three “well-behaved” black hole systems
Cyg X-1, GX 339-4, and GS 1354-644. The solid and dashed lines show the the dependence
Γ (R) expected in the disk-spheroid model and in the plasma ejection model discussed in the text.
Adopted from [17]
Fig. 2.14 R − Γ relation for supermassive black holes, from [67]
2 X-Ray Emission from Black-Hole Binaries
41
the parameters typical for black hole X-ray binaries in the hard spectral state, this
corresponds to the energy range from ∼0.5–1 keV to ∼50–100 keV (e.g., Fig. 2.2).
The photon index Γ of the Comptonized spectrum depends in a rather complicated
way on the parameters of the Comptonizing media, primarily on the electron temperature and the Thompson optical depth [55]. It is more meaningful to relate Γ to the
Comptonization parameter y or, nearly equivalently, to the Compton amplification
factor A. The latter describes the energy balance in the corona and is defined as the
ratio of the energy deposition rate into hot electrons and the energy flux brought into
the Comptonization region by soft seed photons. The concrete shape of the Γ (A)
relation depends on the ratio Tbb /Te of the temperatures of the seed photons and the
electrons, the Thomson optical depth, and the geometry, but broadly speaking, the
higher the Compton amplification factor, the harder is the Comptonized spectrum
[56, 10, 22, 15]. This is illustrated by the results of Monte-Carlo simulations shown
in Fig. 2.15.
Fig. 2.15 Relation between the photon index of Comptonized radiation Γ and the Compton amplification factor A. The symbols show results of Monte-Carlo simulations assuming spherical geometry for different values of parameters of the Comptonization region and soft seed photons. The
solid lines are calculated using Eq. (2.1). Adopted from [17]
The strength of the reflected component in the spectrum depends on the fraction
of the Comptonized radiation intercepted by the accretion disk (Sect. 2.4). The latter
is defined by the geometry of the accretion flow, namely, by the solid angle Ωdisk
subtended by the accretion disk as seen from the corona. In addition, the spectrum
of the reflected emission depends on the ionization state of the disk, in particular, its
low energy part which is formed by the interplay between Thomson scattering and
photoabsorption and fluorescence by metals. The problem is further complicated by
42
M. Gilfanov
the fact that the ionization state of the disk can be modified by the Comptonized
radiation.
Observations show that there is a clear correlation between the photon index
of the Comptonized radiation Γ (i.e., the Comptonization parameter) and the relative amplitude of the reflected component R (Fig. 2.13). Softer spectra (lower
value of the Comptonization parameter y and of the Compton amplification factor
A) have stronger reflected component, revealing itself, for example, via a larger
equivalent width of the iron fluorescent line. The existence of this correlation suggests that there is a positive correlation between the fraction of the Comptonized
radiation intercepted by the accretion disk and the energy flux of the soft seed
photons to the Comptonization region [66]. This is a strong argument in favor of
the accretion disk being the primary source of soft seed photons to the Comptonization region. Indeed, in the absence of strong beaming effects a correlation
between Ωdisk and the seed photons flux should be expected since an increase in
the solid angle of the disk seen by the hot electrons (= Ωdisk ) should generally lead
to the increase of the fraction of the disk emission reaching the Comptonization
region.
2.8.2 Toy Models
We illustrate the above considerations with two simple and idealized models having
a different cause of change of Ωdisk . In the first, the disk-spheroid model (cf. “sombrero” configuration, Sect. 2.2), an optically thin uniform hot sphere with radius
rsph , the source of the hard Comptonized radiation is surrounded by an optically
thick cold disk with an inner radius rdisk , Ωdisk depending on the ratio rdisk /rsph .
Propagation of the the disk toward/inward the hot sphere (decrease of rdisk /rsph )
leads to an increase of the reflection scaling factor R, a decrease of the the Compton amplification factor A, and a steepening of the Comptonized spectrum. In such
a context the model was first studied by Zdziarski et al. [66]. In the second, the
plasma ejection model, the value of the Ωdisk is defined by the intrinsic properties
of the emitting hot plasma, particularly by its bulk motion with mildly relativistic
velocity toward or away from the disk, which itself remains unchanged [5]. In the
case of an infinite disk, values of the reflection scaling factor R below and above
unity correspond to the hot plasma moving, respectively, away from and toward the
disk.
Both models predict a relation between reflection R and Compton amplification
factor A which can be translated to Γ (R) given a dependence Γ (A) of the photon index of the Comptonized spectrum on the amplification factor. The relation
between Γ and A can be approximated by
A = (1 − e−τT ) ·
1 − Γ (Te /Tbb )2−Γ − 1
·
+ e−τT .
2 − Γ (Te Tbb )1−Γ − 1
(2.1)
2 X-Ray Emission from Black-Hole Binaries
43
This formula is based on the representation of the Comptonized spectrum by a power
law in the energy range kTbb − kTe and takes into account that a fraction e−τT of
the soft radiation will leave the Comptonization region unmodified. Despite its simplicity, it agrees with the results of the Monte-Carlo calculations with reasonable
accuracy for optical depth τT ∼ 1 and Tbb /Te ∼ 10−5 − 10−3 (Fig. 2.15).
The expected Γ (R) relations are shown in Fig. 2.13. With a proper tuning of the
parameters, both models can reproduce the observed shape of the Γ (R) dependence
and in this respect are virtually indistinguishable. The models plotted in Fig. 2.13
were calculated with the following parameters: the disk-spheroid model assumes
the disk albedo a = 0.1, Thomson optical depth of the cloud τT = 1, and the ratio
of the temperature of the seed photons to the electron temperature Tbb /Te = 10−4 . I
note that the latter value is too small for stellar mass black holes, a more realistic one
being in the range Tbb /Te = 10−3 . However, this does not invalidate the model as
a number of significant effects are ignored in this calculation which can modify the
Γ (R) dependence, for example, disk inclination, gravitational energy release in the
disk, etc. The exact importance of these factors is yet to be determined. The plasma
ejection model parameters are a = 0.15, τT = 1, Tbb /Te = 3 · 10−3 , and μs = 0.3.
The observed range of the reflection R ∼ 0.3 − 1 and the slope Γ ∼ 1.5 − 2.2
can be explained assuming variation of the disk radius from rdisk ∼ rsph to rdisk ∼ 0
in the disk-spheroid model or variation of the bulk motion velocity from v ∼ 0.4c
away from the disk to v ∼ 0 in the plasma ejection model.
Of course these models are very simple and schematic and the real configuration
of the accretion flow is likely to be far more complex. They are presented here with
the only purpose to demonstrate that simple geometrical considerations can successfully explain the observed correlation between parameters of the Comptonized and
reflected emission in black holes. More sophisticated scenarios are considered, for
example, in [28].
2.8.3 Characteristic Frequencies of Variability
As discussed in Sect. 2.6.3, the power density spectra of accreting black holes above
f ≥ 10−2 Hz have a number of bumps and peaks which define several frequencies
characterizing the variability timescales in the accretion flow. These frequencies
usually correlate with each other, therefore, almost any of them may be used to
represent characteristic variability timescales. As illustrated in Fig. 2.16, a tight correlation exists between the reflection amplitude R and the characteristic variability
frequencies – an increase in the amplitude of the reflected component in the energy
spectrum is accompanied by an increase in the variability frequencies. This correlation covers a remarkably broad dynamical range, nearly two orders of magnitude in
frequency.
Although the precise nature of the characteristic noise frequencies is still unknown,
it is plausible that they are associated with the Keplerian and viscous timescales of
the disk and corona at various characteristic radii, for example, at the truncation
44
M. Gilfanov
Fig. 2.16 The correlations between the amplitude of the reflected component R and the characteristic frequencies of aperiodic variability. The frequency of the second peak in the ν Pν plot
(Fig. 2.8) was used to represent the latter
radius of the disk (Sect. 2.6.3). If this is true, the correlation between R and ν can
be easily understood, at least qualitatively. Indeed, in the truncated disk picture the
increase in reflection is caused by the inward propagation of the inner disk boundary,
hence, it is accompanied by an increase in the Keplerian frequency at the disk truncation radius and the corresponding increase in the characteristic noise frequencies.
2.8.4 Doppler Broadening of the Iron Line
The spectrum of the emission, reflected from a Keplerian accretion disk, is modified
by special and general relativity effects [12], in particular, the width of the fluorescent line of iron is affected by the Doppler effect due to Keplerian motion in the
disk (Sect. 2.4). If the increase in the reflection amplitude is caused by the decrease
in the inner radius of the accretion disk, a correlation should be expected between
the amplitude of reflected emission and its Doppler broadening, in particular, the
Doppler width of the iron line. Such a correlation is a generic prediction of the truncated disk models and might be used to discriminate between different geometries of
the accretion flow. The energy resolution of RXTE/PCA, whose data have been used
for this study [19], is not entirely adequate for the task to accurately measure the
relativistic smearing of the reflection features. However, the data shown in Fig. 2.17
suggest a correlated behavior of the reflection and the Doppler broadening of the
fluorescent line of iron.
2 X-Ray Emission from Black-Hole Binaries
45
Fig. 2.17 Relation between the Doppler broadening of the iron line and the reflection scaling factor
(left) and the characteristic frequency of variability (right). The straight lines in the right panel
1/3
show the dependence ΔE/E ∝ ν Q P O expected in the truncated disk picture if the characteristic
frequencies of variability were proportional to the Keplerian frequency at the inner boundary of
the accretion disk
Speculating further, if the characteristic frequencies of variability are proportional to the Keplerian frequency at the inner boundary of the disk, they should
scale as
−3/2
νQPO ∝ ω K ∝ rdisk .
As the reflected emission is likely to originate primarily from the innermost parts of
the accretion disk, closest to the source of Comptonized radiation, the effect of the
Doppler broadening should be proportional to the Keplerian linear velocity at the
inner edge of the disk:
vK
ΔE
−1/2
∝
sin i ∝ rdisk sin i.
E
c
Therefore, one might expect that the characteristic frequencies of variability and the
Doppler broadening of the fluorescent line should be related via:
νQPO ∝
(ΔE/E)3
,
MBH sin3 i
where MBH is the black hole mass and i is inclination of the binary system. The
PCA data indicate that such dependence might indeed be the case (Fig. 2.17, right
panel). However, an independent confirmation by observations with higher energy
resolution instruments is still needed.
Thus, observations speak in favor of the truncated disk scenario. However,
this point of view is not universally accepted and alternative interpretations and
46
M. Gilfanov
scenarios are being investigated. The counter arguments are based, for example, on
the possible detection of the relativistic broad Fe K line and cool disk emission component in the hard state spectra of several black holes, suggesting that the optically
thick disk might be present in the vicinity of the compact object in the hard state
as well. For the detailed discussion the interested reader is referred to the original
work, e.g., [34] and references therein.
2.9 Comparison with Neutron Star Binaries
Neutron star radii are most likely in the range ∼10–15 km, i.e., of the order
of ∼3–4r g for a ∼1.4M object. This is comparable to the radius of the last
marginally stable Keplerian orbit around a non-rotating black hole; hence, the efficiency η = L X / Ṁc2 of accretion onto a neutron star is not much different from
that on a black hole. Therefore at comparable Ṁ accreting black holes and neutron
stars would have comparable X-ray luminosities (but see below regarding the contribution of the boundary layer). The Eddington luminosity limit, however, is proportional to the mass of the central objects, and the typical maximum luminosity is
by M B H /M N S ∼ 5–10 times smaller for neutron stars. This is in general agreement
with observations of peak luminosities of black hole transients and their comparison
to luminosities of persistent and transient neutron star systems in the Milky Way.
The fact that the size of a neutron star is of the order of ∼3r g , i.e., is comparable
to the radius of the last marginally stable Keplerian orbit around a black hole also
suggests that the structure of the accretion disk in both cases may be similar. This
is indirectly confirmed by the existence of two spectral states in accreting neutron
stars, whose properties are qualitatively similar to black holes (Fig. 2.1). A more
direct argument is presented by the similarity of the accretion disk spectra in the
soft state, which can be described by the same spectral models in black holes and
neutron stars (e.g. [20]). Moreover, the power density spectra of the accretion disk
in both types of systems follow a Pν ∝ ν −α power law with α close to ∼1 (Sect. 2.6
and [20]). The strong magnetic field which may exist around young neutron stars
can change the picture, causing disruption of the accretion disk at a large distance
from the neutron star and modifying dramatically the structure of the accretion flow
inside the magnetospheric radius rm r N S . This may lead to the phenomenon of
X-ray pulsations common in high-mass X-ray binaries and is not considered in this
chapter.
The main qualitative difference between the two types of compact objects is
obviously the existence of a solid surface of the neutron star which is absent in
the case of a black hole. Neutron star rotation frequencies are typically in the few
hundred Hz range, i.e., ∼ several times smaller than the Keplerian frequency near
its surface, ν K ∼ kHz. Therefore, a boundary or spreading layer will appear near
the surface of the neutron star where accreting matter decelerates from Keplerian
rotation in the accretion disk down to the neutron star spin frequency and settles onto
its surface (Fig. 2.18, left panels). In Newtonian approximation, half of the energy of
the test particle on a Keplerian orbit is in the form of kinetic energy of the Keplerian
2 X-Ray Emission from Black-Hole Binaries
47
Fig. 2.18 The left panels show the geometry of the spreading layer on the surface of the neutron
star (upper) and radial dependence of the angular velocity of accreting matter (lower). Adopted
from [24]. On the right: Hard state spectra of several weakly magnetized accreting neutron stars
and of a black hole Cyg X-1. Based on the data from RXTE observations
rotation; hence, for a non-rotating neutron star half of the energy release due to
accretion would take place in the boundary/spreading layer. The effects of general
relativity can increase this fraction, e.g., up to ∼2/3 in the case of a neutron star
with radius r N S = 3r g [52, 50]. Rotation of the neutron star and deviations of the
space–time geometry from Schwarzschild metric further modify the fraction of the
energy released on the star’s surface. A luminous spectral component emitted by
the boundary layer will exist in the X-ray spectrum of an accreting neutron star, in
addition to the emission from the accretion disk. The shape of the spectra of luminous neutron stars (soft state spectrum in Fig. 2.1) suggests that both the accretion
disk and the boundary layer are in the optically thick regime. As the luminosities of
the two components are comparable, but the emitting area of the boundary layer is
smaller than that of the accretion disk, its spectrum is expected to be harder.
Looking from a different angle we may say that because of the presence of the
stellar surface, at the same Ṁ neutron stars are approximately twice more luminous
than black holes. Indeed, in the case of the black hole accretion the kinetic energy of
Keplerian motion at the inner edge of the accretion disk is nearly all advected into
the black hole. If the central object is a neutron star, this energy is released in the
boundary/spreading layer on its surface, approximately doubling the luminosity of
the source.1 An interesting consequence of this is that the ratio of L Edd /M is larger
for neutron stars than for black holes.
1 A more accurate consideration should take into account geometry of the problem, namely the
emission diagrams and the orientation with respect to the observer of the emitting surfaces of the
boundary layer and accretion disk. This makes the boundary layer contribution to the observed
emission dependent on the inclination of the binary system, see, for example, [20].
48
M. Gilfanov
Due to the similarity of the spectra of the accretion disk and the boundary layer,
the total spectrum has a smooth curved shape, which is difficult to decompose into
separate spectral components. This complicates analysis and interpretation of the
neutron star spectra and in spite of a very significant increase in the sensitivity of
X-ray instruments made in the last decade, still often leads to ambiguous and contradicting results, even based on physically motivated spectral models. This degeneracy
may be removed with the help of timing information. It has been noticed already in
the early 1980s that the variability patterns may be different for the boundary layer
and accretion disk [35]. Further progress has been made almost 20 years later, thanks
to the large collecting area of the PCA instrument aboard RXTE and the use of novel
data analysis techniques. Using the method of frequency-resolved spectral analysis
it has been shown that aperiodic and quasi-periodic variability of bright LMXBs –
atoll and Z-sources, on ∼ s – ms timescales is caused primarily by variations of the
luminosity of the boundary layer [20]. It was also shown that the boundary layer
spectrum remains nearly constant in the course of the luminosity variations and
is represented to certain accuracy by the Fourier frequency-resolved spectrum. This
permits to resolve the degeneracy in the spectral fitting and to separate contributions
of the boundary layer and disk emission (Fig. 2.19, left panel). Interestingly, the
spectrum of the boundary layer emission has the same shape in different objects and
Fig. 2.19 Left: Points with error bars showing total and frequency-resolved spectra of GX 340+0.
The dashed (blue in the color version of the plot) and upper solid (red) histograms show the disk
and boundary layer spectra, the latter computed as a difference between the (observed) total and
(predicted) accretion disk spectrum. The lower solid histogram is obtained from the upper scaling
it to the total energy flux of the frequency-resolved spectrum. Adopted from [20]. Right: Fourier
frequency-resolved spectra, corrected for the interstellar absorption (≈boundary layer spectra) of
five luminous accreting neutron stars. For 4U 1608-52, the frequency-resolved spectrum of the
lower kHz QPO is shown. All spectra were corrected for the interstellar absorption. The thick
short-dashed line shows the best fit Comptonization model with kTs = 1.5, kTe = 3.3 keV, τ = 5.
The thin long-dashed line shows a black body spectrum with temperature kTbb = 2.4 keV. Adopted
from [45]
2 X-Ray Emission from Black-Hole Binaries
49
is nearly independent of the global mass accretion rate in the investigated range of
Ṁ ∼ (0.1 − 1) ṀEdd and in the limit of Ṁ ∼ ṀEdd is close to a Wien spectrum
with kT ∼ 2.4 keV (Fig. 2.19, right panel). Its independence on the global value
of Ṁ lends support to the theoretical suggestion by [24] that the boundary layer
is radiation pressure supported. With this assumption, one can attempt to measure
gravity on the neutron star surface and hence M/R 2 , from the shape of the boundary
layer spectrum, similarly to the photospheric expansion (i.e., Eddington limited) Xray bursts. This gives results within the range of values obtained by other methods
[45, 51].
As neutron stars have smaller mass than black holes, the linear scale corresponding to the gravitational radius is smaller, r g ∝ M. This has two important consequences. First, the surface area of the emitting region is directly proportional to M 2 ,
therefore at the same luminosity the temperature of black body emission will be
larger, Tbb ∝ M −1/2 . The spectrum is further modified by the contribution of the
emission from the boundary layer, which has yet smaller emitting region and harder
spectrum. This is illustrated by Fig. 2.1 – the soft state spectrum of the neutron star
4U 1705-44 is noticeably harder than that of the black hole Cyg X-1.
Second, the smaller linear scale in neutron star accretion shifts the characteristic
frequency scales by a factor ∝ M −1 , suggesting ∼5 times higher frequencies of variability in neutron star systems. This is confirmed by observations as demonstrated
in the extensive comparison of black hole and neutron star power density spectra
in [59]. In addition, characteristic timescales in the boundary or spreading layer
are significantly shorter than those in the accretion disk and are in the tens of kHz
frequency domain. It has been suggested that a very high frequency component may
exist in the power density spectra of neutron stars associated with the turbulence in
the spreading layer [59, 24]. If detected, this may become a unique diagnostics tool
of physical conditions in the spreading layer on the surface of the neutron star. So
far, only upper limits have been obtained; they are at the level of ∼10−2 fractional
rms [59].
The relatively cold surface of the neutron star is a source of copious soft photons.
This is mostly relevant in the hard spectral state – the low energy photons emitted by
the neutron star surface result in a more efficient cooling of electrons in the Comptonization region, thus changing its energy balance and Comptonization parameter
(luminosity enhancement factor, cf. Sect. 2.8.1). Consequently, the energy spectra
of neutron stars in the hard state are significantly softer than those of black holes –
they have larger spectral index (smaller Comptonization parameter) and a smaller
value of the high energy cut-off (lower electron temperature). Both these effects are
clearly seen in the spectra shown in Figs. 2.1, 2.18.
The existence of soft and hard spectral states in neutron stars suggests that, similarly to black holes, the accretion flow can change its configuration from optically
thin to optically thick. A remarkable fact is that this change seems to occur to both
accretion disk and boundary layer (quasi-) simultaneously. Indeed, to my knowledge, no two-component (soft + hard) spectrum has been observed in the case of
accreting neutron stars so far.
50
M. Gilfanov
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